Finite Group Databasebeta

← #28401375

Subgroups — #28401375

Order 14464.

462 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×D113C211yesmaximal, characteristic, complemented
2#121×D113C211yesmaximal, characteristic
3#20298935C211yesmaximal, characteristic, complemented
4#20298932C211yesmaximal, characteristic, complemented
5#20298913C211yesmaximal, characteristic, complemented
6#20298910C211yesmaximal, characteristic, complemented
7#20298846C211yesmaximal, characteristic, complemented
8#20298844C211yesmaximal, characteristic, complemented
9#20298842C211yesmaximal, characteristic, complemented
10#20298750C211yesmaximal, characteristic, complemented
11#20298747C211yesmaximal, characteristic, complemented
12#20298738C211yesmaximal, characteristic, complemented
13#20298735C211yesmaximal, characteristic, complemented
14#20298561C211yesmaximal, characteristic, complemented
15#468×C113C211yesmaximal, characteristic, complemented
16C2×Q8×D113C2211yescharacteristic, complemented
17C2×D4×D113—21no—
18C2×D4×D113C2211yescharacteristic
19C22×C4×D113C2211yescharacteristic
20C2×#14268636C2211yescharacteristic, complemented
21C2×#14268635C2211yescharacteristic, complemented
22C2×#14268634C2211yescharacteristic, complemented
23C2×#14268631C2211yescharacteristic, complemented
24#16813602C2211yescharacteristic
25#16813601C2211yescharacteristic, complemented
26#16813600—21no—
27#16813599—21no—
28#16813598C2211yescharacteristic
29#16813597C2211yescharacteristic
30#16813596C2211yescharacteristic
31#16813595—21no—
32#16813594—21no—
33#16813593C2211yescharacteristic
34C4⋊C4×D113—21no—
35C4⋊C4×D113C2211yescharacteristic
36C4⋊C4×D113C2211yescharacteristic, complemented
37C22⋊C4×D113—21no—
38C22⋊C4×D113—21no—
39#16813556C2211yescharacteristic, complemented
40#16813555C2211yescharacteristic, complemented
41#16813555C2211yescharacteristic, complemented
42#16813547C2211yescharacteristic, complemented
43#16813546C2211yescharacteristic, complemented
44#16813545C2211yescharacteristic, complemented
45#16813544C2211yescharacteristic, complemented
46#16813543C2211yescharacteristic, complemented
47#16813542C2211yescharacteristic, complemented
48#16813534C2211yescharacteristic
49#16813534C2211yescharacteristic
50#16813534C2211yescharacteristic, complemented
51#16813534C2211yescharacteristic, complemented
52C2×SD16×C113C2211yescharacteristic, complemented
53C2×M16×C113C2211yescharacteristic, complemented
54C22⋊Q8×C113C2211yescharacteristic, complemented
55#121×C113C2211yescharacteristic
56#107×C113C2211yescharacteristic, complemented
57#103×C113C2211yescharacteristic, complemented
58#102×C113C2211yescharacteristic, complemented
59C23×D113—21no—
60C23×D113D411yescharacteristic
61C2×Dic226—21no—
62C2×Dic226C2311yescharacteristic
63Q8×D113—41no—
64Q8×D113—41no—
65C2×D452—21no—
66C2×D452C2311yescharacteristic
67C2×C22⋊D113—21no—
68C2×C22⋊D113—21no—
69C2×C22⋊D113—21no—
70D4×D113—41no—
71D4×D113—41no—
72D4×D113—44no—
73C22×Dic113D411yescharacteristic
74C2×C4×D113—21no—
75C2×C4×D113—21no—
76C2×C4×D113—21no—
77C2×C4×D113—41no—
78C2×C4×D113—41no—
79C2×C4×D113D411yescharacteristic
80C2×C4×D113D411yescharacteristic
81C2×C4×D113C2311yescharacteristic
82C2×C4×D113C2311yescharacteristic
83#14268636—41no—
84#14268635—41no—
85#14268634—41no—
86#14268631—41no—
87C2×#11744751C2311yescharacteristic
88C2×#11744751C2311yescharacteristic
89C4⋊Dic113—21no—
90C4⋊Dic113C2311yescharacteristic
91C4⋊Dic113C2311yescharacteristic
92#14268626—21no—
93#14268626—21no—
94#14268626—21no—
95#14268626—21no—
96C22⋊Dic113—21no—
97C22⋊Dic113—21no—
98#14268624—21no—
99#14268624—21no—
100#14268624—21no—
101#14268624—21no—
102C2×Q8×C113C2311yescharacteristic
103C2×D4×C113—21no—
104C2×D4×C113C2311yescharacteristic
105C22×C4×C113C2311yescharacteristic
106SD16×C113—41no—
107M16×C113—41no—
108C2×C8×C113C2311yescharacteristic
109C2×C8×C113C2311yescharacteristic
110C4⋊C4×C113—21no—
111C4⋊C4×C113C2311yescharacteristic
112C4⋊C4×C113C2311yescharacteristic
113C22⋊C4×C113—21no—
114C22⋊C4×C113—21no—
115C23×C113—21no—
116C23×C113C2×D411yescharacteristic
117C22×D113—22no—
118C22×D113—22no—
119C22×D113—21no—
120C22×D113—41no—
121C22×D113—42no—
122C22×D113—42no—
123C22×D113—42no—
124C22×D113C2×D411yescharacteristic
125C22×D113C2×D411yescharacteristic
126Q8×C113—21no—
127Q8×C113—21no—
128D4×C113—21no—
129D4×C113—21no—
130D4×C113—42no—
131C2×C4×C113—21no—
132C2×C4×C113—21no—
133C2×C4×C113—21no—
134C2×C4×C113—21no—
135C2×C4×C113—21no—
136C2×C4×C113C2×D411yescharacteristic
137C2×C4×C113C2411yescharacteristic
138C22⋊D113—42no—
139C22⋊D113—42no—
140C22⋊D113—42no—
141C2×Dic113—21no—
142C2×Dic113—21no—
143C2×Dic113—21no—
144C2×Dic113—41no—
145C2×Dic113C2×D411yescharacteristic
146C2×Dic113C2×D411yescharacteristic
147D452—21no—
148D452—21no—
149D452—42no—
150C4×D113—21no—
151C4×D113—21no—
152C4×D113—21no—
153C4×D113—21no—
154C4×D113—41no—
155C4×D113—41no—
156C4×D113—42no—
157C4×D113—42no—
158C4×D113—42no—
159Dic226—21no—
160Dic226—21no—
161Dic226—42no—
162C8×C113—21no—
163C8×C113—21no—
164#11744751—21no—
165#11744751—21no—
166C22×C113—21no—
167C22×C113—21no—
168C22×C113—21no—
169C22×C113—41no—
170C22×C113—41no—
171C22×C113—41no—
172C22×C113C22×D411yescharacteristic
173C2×D113—21no—
174C2×D113—21no—
175C2×D113—21no—
176C2×D113—21no—
177C2×D113—44no—
178C2×D113—41no—
179C2×D113—41no—
180C2×D113—41no—
181C2×D113—84no—
182C2×D113AGL(1,ℤ/8)12yes—
183C2×D113#13712yes—
184C4×C113—21no—
185C4×C113—41no—
186C4×C113—41no—
187C4×C113—41no—
188C4×C113H(5,𝔽2)11yescharacteristic
189C4×C113H(5,𝔽2)11yescharacteristic
190Dic113—21no—
191Dic113—21no—
192Dic113—41no—
193Dic113—41no—
194Dic113—41no—
195C2×C113—41no—
196C2×C113—81no—
197C2×C113#53411yescharacteristic
198C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
199C2×C113C2×#13711yescharacteristic
200D113—22no—
201D113—22no—
202D113—81no—
203#3193—1131nomaximal, Sylow
204C113#319311yescharacteristic, complemented, Sylow
205C2×C22⋊Q8—1131no—
206C2×#121—1131no—
207#484—1131no—
208#484—1131no—
209#480—1131no—
210#480—1131no—
211#468—1131no—
212#468—1131no—
213#468—1131no—
214#468D11311yescharacteristic, complemented
215#450—1131no—
216#450—1131no—
217#448—1131no—
218#448—1131no—
219#407—1131no—
220C22×Q8—1131no—
221C22×D4—1131no—
222C22×D4—2261no—
223C23×C4—1131no—
224C2×SD16—1131no—
225C2×SD16—1131no—
226C2×SD16—1131no—
227C2×SD16C2×D11311yescharacteristic, complemented
228C2×M16—1131no—
229C2×M16C2×D11311yescharacteristic, complemented
230C22⋊Q8—1131no—
231C22⋊Q8—1131no—
232C22⋊Q8—1131no—
233C22⋊Q8—2261no—
234C22⋊Q8—2261no—
235C22⋊Q8C2×D11311yescharacteristic, complemented
236#121—1131no—
237#121—1131no—
238#121—1131no—
239#121—2261no—
240#121—2261no—
241#121C2×D11311yescharacteristic
242C2×C4⋊C4—1131no—
243C2×C4⋊C4—1131no—
244C2×C4⋊C4—2261no—
245C2×C22⋊C4—2261no—
246C2×C22⋊C4—2261no—
247#107—1131no—
248#107—1131no—
249#107—1131no—
250#107C2×D11311yescharacteristic, complemented
251#103—1131no—
252#103—1131no—
253#103—1131no—
254#103C2×D11311yescharacteristic, complemented
255#102—1131no—
256#102—1131no—
257#102—1131no—
258#102C2×D11311yescharacteristic, complemented
259C22⋊C8—1131no—
260C22⋊C8—1131no—
261C22⋊C8—1131no—
262C22⋊C8—1131no—
263C24—1131no—
264C24—2261no—
265C2×Q8—1131no—
266C2×Q8—4521no—
267C2×Q8—4521no—
268C2×Q8—2261no—
269C2×Q8C22×D11311yescharacteristic
270C2×D4—21no—
271C2×D4—1131no—
272C2×D4—4524no—
273C2×D4—4521no—
274C2×D4—4521no—
275C2×D4—2261no—
276C2×D4—2261no—
277C2×D4—2261no—
278C2×D4—2261no—
279C2×D4C22×D11311yescharacteristic
280C22×C4—1131no—
281C22×C4—1131no—
282C22×C4—1131no—
283C22×C4—1131no—
284C22×C4—1131no—
285C22×C4—4521no—
286C22×C4—4521no—
287C22×C4—2261no—
288C22×C4—2261no—
289C22×C4—2261no—
290C22×C4C22×D11311yescharacteristic
291SD16—41no—
292SD16—4521no—
293SD16—4521no—
294SD16—4521no—
295M16—41no—
296M16—4521no—
297C2×C8—1131no—
298C2×C8—1131no—
299C2×C8C22×D11311yescharacteristic
300C2×C8C22×D11311yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—1131no—
303C4⋊C4—1131no—
304C4⋊C4—2261no—
305C4⋊C4—2261no—
306C4⋊C4—2261no—
307C4⋊C4—2261no—
308C4⋊C4—2261no—
309C4⋊C4C22×D11311yescharacteristic
310C4⋊C4C22×D11311yescharacteristic
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—2261no—
314C22⋊C4—2261no—
315C22⋊C4—2261no—
316C22⋊C4—2261no—
317C22⋊C4—2261no—
318C22⋊C4—2261no—
319C23—21no—
320C23—1131no—
321C23—1131no—
322C23—4521no—
323C23—4522no—
324C23—4522no—
325C23—4522no—
326C23—2262no—
327C23—2262no—
328C23—2261no—
329C23D4×D11311yescharacteristic
330Q8—21no—
331Q8—21no—
332Q8—4522no—
333Q8—2261no—
334Q8—2261no—
335D4—21no—
336D4—21no—
337D4—42no—
338D4—4522no—
339D4—4522no—
340D4—4522no—
341D4—4522no—
342D4—2261no—
343D4—2261no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—1131no—
350C2×C4—1131no—
351C2×C4—4521no—
352C2×C4—4521no—
353C2×C4—4522no—
354C2×C4—4522no—
355C2×C4—4522no—
356C2×C4—4521no—
357C2×C4—2261no—
358C2×C4—2261no—
359C2×C4—2261no—
360C2×C4—2261no—
361C2×C4—2261no—
362C2×C4—2261no—
363C2×C4—2261no—
364C2×C4D4×D11311yescharacteristic
365C2×C4C23×D11311yescharacteristic
366C8—21no—
367C8—21no—
368C8—2261no—
369C8—2261no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—1132no—
377C22—1132no—
378C22—9044no—
379C22—4524no—
380C22—4521no—
381C22—4521no—
382C22—4521no—
383C22—2261no—
384C22—2261no—
385C22—2261no—
386C22—2261no—
387C22C2×D4×D11311yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—4521no—
393C4—4521no—
394C4—4521no—
395C4—2261no—
396C4—2261no—
397C4#1681366711yescharacteristic
398C4#1681366811yescharacteristic
399C2—41no—
400C2—81no—
401C2—9041no—
402C2—2262no—
403C2—2262no—
404C2#2029915611yescharacteristic
405C2AGL(1,ℤ/8)×D11311yescharacteristic
406C2#137×D11311yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#468×C113C2
socleC22×C113C22×D4
prime radical#121×D113C2
hypercenter#468D113
hypocenterC113#3193
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22⋊Q8×D113C2semidirect
3. #20298935C2semidirect
4. #20298932C2semidirect
5. #20298913C2semidirect
6. #20298910C2semidirect
7. #20298846C2semidirect
8. #20298844C2semidirect
9. #20298842C2semidirect
10. #20298750C2semidirect
11. #20298747C2semidirect
12. #20298738C2semidirect
13. #20298735C2semidirect
14. #20298561C2semidirect
15. #468×C113C2semidirect
16. C2×Q8×D113C22semidirect
20. C2×#14268636C22semidirect
21. C2×#14268635C22semidirect
22. C2×#14268634C22semidirect
23. C2×#14268631C22semidirect
25. #16813601C22semidirect
36. C4⋊C4×D113C22semidirect
39. #16813556C22semidirect
40. #16813555C22semidirect
41. #16813555C22semidirect
42. #16813547C22semidirect
43. #16813546C22semidirect
44. #16813545C22semidirect
45. #16813544C22semidirect
46. #16813543C22semidirect
47. #16813542C22semidirect
50. #16813534C22semidirect
51. #16813534C22semidirect
52. C2×SD16×C113C22semidirect
53. C2×M16×C113C22semidirect
54. C22⋊Q8×C113C22semidirect
56. #107×C113C22semidirect
57. #103×C113C22semidirect
58. #102×C113C22semidirect
204. C113#3193semidirect
214. #468D113semidirect
227. C2×SD16C2×D113semidirect
229. C2×M16C2×D113semidirect
235. C22⋊Q8C2×D113semidirect
250. #107C2×D113semidirect
254. #103C2×D113semidirect
258. #102C2×D113semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22⋊Q8×D1132
#121×D1132
#202989352
#202989322
#202989132
#202989102
#202988462
#202988442
#202988422
#202987502
#202987472
#202987382
#202987352
#202985612
#468×C1132
#3193113